Abstract
Abstract
This chapter studies Cauchy’s Formula and its applications. Cauchy’s Formula says that the values of f on L rigidly determine its values everywhere inside L. The chapter offers two explanations, both of which are firmly rooted in Cauchy’s Theorem. It then looks at infinite differentiability and Taylor series. Returning to the case where L is a simple loop, the chapter shows that if f(z) is analytic inside L then so is f1(z). From this it will follow by induction that f(z) is infinitely differentiable. The chapter also considers the calculus of residues and the annular Laurent series, which is the natural generalization of the Taylor series when the centre of the expansion is a pole rather than a non-singular point.
Publisher
Oxford University PressOxford
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